The present study investigates whether second language (L2) speakers are sensitive to the information-structural constraints and frequency distributions at the syntax–discourse interface in the L2. L1-German–L2-English and L1-English–L2-German speakers completed a speeded naturalness judgment task. For sentences presented in broad or narrow-focus contexts, they judged the naturalness of fronted locative (LP) and temporal (TP) adverbial phrases and fronted objects in both English and German. English and German differ in the frequency with which they employ these constructions. With high-frequency fronted-LP and TP sentences, both the L2 English and L2 German speakers exhibited equivalent judgments as their L1 counterparts, in spite of differences in the perceived naturalness and relative frequency of these constructions in English vs. German. Like L1 speakers, L2 English and L2 German speakers also judged the less-frequent fronted objects as more natural in narrow-focus than broad-focus contexts, showing successful acquisition at the syntax–discourse interface. However, they judged fronted object sentences as more natural overall than their respective L1 counterparts in both English and German. Together, these findings suggest that convergence at the L2 syntax–discourse interface is possible per se, but that lower construction frequency in the input entails persistent overgeneralization of non-canonical options in the L2.